login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A276323 a(n) = (binomial(2 * prime(n + 3), prime(n + 3)) * A005259(prime(n + 3) - 1) - 2)/prime(n + 3)^5 for n >= 1. 2
4382314, 59821998476834, 338197165389273486, 17314015796594772560245514, 145853326344012138627669357202, 12936469013977571458378002436843685186, 15931675838688077485749893663903436780403973163302 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Let p be a prime > 5. Binomial(2 * p, p) * A005259(p - 1) == 2 (mod p^5). So a(n) is an integer.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..88

Julian Rosen, Periods, supercongruences, and their motivic lifts, arXiv:1608.06864 [math.NT], 2016.

EXAMPLE

a(1) = (binomial(14, 7) * A005259(6) - 2)/7^5 = (3432 * 21460825 - 2)/7^5 = 4382314.

MATHEMATICA

Table[(Binomial[2 Prime[n + 3], Prime[n + 3]] Sum[(Binomial[#, k] Binomial[# + k, k])^2, {k, 0, #}] &[Prime[n + 3] - 1] - 2)/Prime[n + 3]^5, {n, 7}] (* Michael De Vlieger, Aug 30 2016 *)

PROG

(Ruby)

require 'prime'

def C(n, r)

  r = [r, n - r].min

  return 1 if r == 0

  return n if r == 1

  numerator = (n - r + 1..n).to_a

  denominator = (1..r).to_a

  (2..r).each{|p|

    pivot = denominator[p - 1]

    if pivot > 1

      offset = (n - r) % p

      (p - 1).step(r - 1, p){|k|

        numerator[k - offset] /= pivot

        denominator[k] /= pivot

      }

    end

  }

  result = 1

  (0..r - 1).each{|k|

    result *= numerator[k] if numerator[k] > 1

  }

  return result

end

def A005259(n)

  i = 0

  a, b = 1, 5

  ary = [1]

  while i < n

    i += 1

    a, b = b, ((((34 * i + 51) * i + 27) * i + 5) * b - i ** 3 * a) / (i + 1) ** 3

    ary << a

  end

  ary

end

def A276323(n)

  p_ary = Prime.take(n + 3)[3..-1]

  a = A005259(p_ary[-1] - 1)

  ary = []

  p_ary.each{|i|

    j = C(2 * i, i) * a[i - 1] - 2

    break if j % i ** 5 > 0

    ary << j / i ** 5

  }

  ary

end

CROSSREFS

Cf. A000984, A005259.

Sequence in context: A254259 A254305 A237537 * A288086 A210297 A019288

Adjacent sequences:  A276320 A276321 A276322 * A276324 A276325 A276326

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Aug 30 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 22 02:46 EST 2022. Contains 350481 sequences. (Running on oeis4.)